

A343400


Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.


0



1, 2, 3, 4, 4, 5, 4, 6, 4, 9, 5, 7, 8, 6, 9, 6, 7, 9, 7, 6, 9, 7, 8, 7, 7, 10, 6, 9, 11, 9, 12, 8, 9, 14, 5, 13, 11, 8, 11, 11, 7, 13, 9, 12, 11, 9, 9, 11, 8, 12, 11, 11, 11, 6, 16, 4, 11, 12, 11, 13, 12, 6, 10, 9, 8, 17, 8, 12, 11, 10, 8, 10, 12, 10, 8, 11, 12, 12, 13, 7
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OFFSET

1,2


COMMENTS

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..2*10^6.
The first indices n for which a(n) = 0 are 4051736, 7479656, 8592680, 9712160, 14039792, 16726256, 24914510.  Giovanni Resta, Apr 14 2021


LINKS



EXAMPLE

a(2) = 2 with 2 = 3^0 + [1^2/3] + [2^2/4] = 3^0 + [2^2/3] + [1^2/4].
a(2942) = 2 with 2942 = 3^1 + [93^2/3] + [15^2/4] = 3^7 + [44^2/3] + [21^2/4].
a(627662) = 5 with 627662  3^0 = [330^2/3] + [1538^2/4] = [1042^2/3] + [1031^2/4] = [1318^2/3] + [441^2/4] = [1328^2/3] + [399^2/4] = [1352^2/3] + [271^2/4].
a(1103096) = 3 with 1103096 = 3^1 + [260^2/3] + [2079^2/4] = 3^1 + [508^2/3] + [2017^2/4] = 3^9 + [328^2/3] + [2047^2/4].
a(1694294) = 3 with 1694294 = 3^8 + [860^2/3] + [2401^2/4] = 3^8 + [928^2/3] + [2367^2/4] = 3^13 + [112^2/3] + [619^2/4].


MATHEMATICA

PowQ[n_]:=PowQ[n]=IntegerQ[Log[3, n]];
tab={}; Do[r=0; Do[If[PowQ[nFloor[x^2/3]Floor[y^2/4]], r=r+1], {x, 1, Sqrt[3n1]}, {y, 1, Sqrt[4(nFloor[x^2/3]1)+1]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



